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Exactly solvable Stuart-Landau models in arbitrary dimensions

Pragjyotish Bhuyan Gogoi, Rahul Ghosh, Debashis Ghoshal, Awadhesh Prasad, Ram Ramaswamy

Abstract

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions $D >2$ and give an exact solution of the oscillator equations in the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has $N=\lfloor{D/2}\rfloor$ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd $D$ there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere $\mathbb{S}^{D-1}$ and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits each of which has the geometry of a torus $\mathbb{T}^N$ on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalisations.

Exactly solvable Stuart-Landau models in arbitrary dimensions

Abstract

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions and give an exact solution of the oscillator equations in the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits each of which has the geometry of a torus on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalisations.

Paper Structure

This paper contains 18 equations, 3 figures.

Figures (3)

  • Figure 1: (Color online) The $D=3$ Stuart-Landau system with parameters $(\mu_{12}, \mu_{23}, \mu_{13}) = (-0.3,-0.2,0.1)$. For any initial ${\mathbf x}(0)$ (marked by a red dot) on the cone the system described by Eq. \ref{['eq:12']} asymptotes to the circle $\mathbb{S}^1$ (solid blue line) at which the cone intersects the sphere $\mathbb{S}^2$. The normal mode coordinate $y_3$ (red) is along $\hat{{\mathbf n}}$, the axis of the cone, while $y_1, y_2$ span the plane perpendicular to it. The initial bivector-plane $\mathsf{P}_2(0)$ (shown in blue) intersects $\mathbb{S}^2$ in a great circle, $\mathcal{C}(0)$ (dashed black line). The ray from the origin to ${\mathbf x}(0)$ intersects $\mathcal{C}(0)$ and $\mathbb{S}^1$ at a common point marked as a black dot.
  • Figure 2: Trajectories in the $D=4$ Stuart-Landau system for typical values of $\mu_{ij}$ are quasiperiodic; a projection is shown in (a). If the ratio of mode frequencies is rational then the orbit is a closed loop as shown in (b) when $\omega_1/\omega_2$ = 2.
  • Figure 3: (Color online) The solid blue and dashed red lines are projections of two trajectories for the four-dimensional system with quaternionic truncation as described in the text. The parameter values are $(\nu_1, \nu_2, \nu_3) = (-1.6, 0.3, -0.2)$ for both orbits which have different initial conditions.